PROPOSITION C. THEORF If two triangles have the three sides three sides of the other, each to each, th in all respects. OPOSITION C [Book I. Let the three sides to each, that is, AB Then must the t Imagine the ▲ ABC, in su vertex D fal' A falls; a CASE T be the given angle. required to bisect any pt. D. BAC. In AC make AE-AD, and join DE. On DE, on the side remote from A, describe an quilat. ADFE. Join AF. Then AF will bisect BAC. For in as AFD, AFE, AD=AE, and AF is common, and FD=FE, that is, BAC is bisected by AF. I. 1. I. c. Q. E. F. Ex. 1. Shew that we can prove this Proposition by means of Prop. IV. and PROP. A., without applying Prop. C. Ex. 2. If the equilateral triangle, employed in the construction, be described with its vertex towards the given angle; shew that there is one case in which the construction will fail, and two in which it will hold good. NOTE.-The line dividing an angle into two equal parts is Called the BISECTOR of the angle. Bisect ▲ ACB by the st. line CD meeting AB in D; then AB shall be bisected in D. For in as ACD, BCD, itting AB · AC=BC, and CD is common, and ▲ ACD= ▲ BCD, .. AD=BD ; .. AB is bisected in D. I. 10. I. 4. Q. E. F. Ex. 1. The straight line, drawn to bisect the vertical angle of an isosceles triangle, also bisects the base. Ex. 2. The straight line, drawn from the vertex of an isosceles triangle to bisect the base, also bisects the vertical angle. Ex. 3. Produce a given finite straight line to a point, such that the part produced may be one-third of the line, which is made up of the whole and the part produced. PROPOSITION IX. PROBLEM. To bisect a given angle. A B Let BAC be the given angle. In AB take any pt. D. In AC make AE=AD, and join DE. On DE, on the side remote from A, describe an equilat. A DFE. Join AF. Then AF will bisect 4 BAC. For in As AFD, AFE, · AD=AE, and AF is common, and FD=FE, that is, BAC is bisected by AF. I. 1. I. c. Q. E. F. Ex. 1. Shew that we can prove this Proposition by means of Prop. IV. and PROP. A., without applying Prop. C. Ex. 2. If the equilateral triangle, employed in the construction, be described with its vertex towards the given angle; shew that there is one case in which the construction will fail, and two in which it will hold good. NOTE.-The line dividing an angle into two equal parts is called the BISECTOR of the angle. PROPOSITION X. PROBLEM. To bisect a given finite straight line. A Let AB be the given st. line. On AB describe an equilat. ▲ ACB. I. 1. Bisect ACB by the st. line CD meeting AB in D; I.9. then AB shall be bisected in D. For in as ACD, BCD, ·· AC=BC, and CD is common, and ▲ ACD= 2 BCD, .. AD=BD; .. AB is bisected in D. I. 4. Q. E. F. Ex. 1. The straight line, drawn to bisect the vertical angle of an isosceles triangle, also bisects the base. Ex. 2. The straight line, drawn from the vertex of an isosceles triangle to bisect the base, also bisects the vertical angle. Ex. 3. Produce a given finite straight line to a point, such that the part produced may be one-third of the line, which is made up of the whole and the part produced. PROPOSITION XI. PROBLEM. To draw a straight line at right angles to a given straight line from a given point in the same. Let AB be the given st. line, and C a given pt. in it. Take any pt. D in AC, and in CB make CE=CD. Join FC. FC shall be 1 to AB. I. 1. For in As DCF, ECF, :: DC=CE, and CF is common, and FD=FE, .. 4 DCFL ECF; and .. FC is 1 to AB. I. c. Def. 9. Q. E. F. COR. To draw a straight line at right angles to a given straight line AC from one extremity, C, take any point D in AC, produce AC to E, making CE-CD, and proceed as in the proposition. Ex. 1. Shew that in the diagram of Prop. ix. AF and ED intersect each other at right angles, and that ED is bisected by AF. Ex. 2. If O be the point in which two lines, bisecting AB and AC, two sides of an equilateral triangle, at right angles, meet; shew that OA, OB, OC are all equal. Ex. 3. Shew that Prop. XI. is a particular case of Prop. IX. |